Back to your sentence, my understanding of it is that the person referred to was trying to speak in a voice that appeared to be authoritative. "He" did this to make others respect and obey "him".The 1995 Longman Dictionary also lists the idiom "the voice of authority" and explains that "it means a way of speaking that makes people respect you." Your sentence is a good example in this sense.
Hope my explanation helps.
Student or Learner